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Social recommendation has achieved great success in many domains including e-commerce and location-based social networks. Existing methods usually explore the user-item interactions or user-user connections to predict users’ preference behaviors. However, they usually learn both user and item representations in Euclidean space, which has large limitations for exploring the latent hierarchical property in the data. In this article, we study a novel problem of hyperbolic social recommendation, where we aim to learn the compact but strong representations for both users and items. Meanwhile, this work also addresses two critical domain-issues, which are under-explored. First, users often make trade-offs with multiple underlying aspect factors to make decisions during their interactions with items. Second, users generally build connections with others in terms of different aspects, which produces different influences with aspects in social network. To this end, we propose a novel graph neural network (GNN) framework with multiple aspect learning, namely, HyperSoRec. Specifically, we first embed all users, items, and aspects into hyperbolic space with superior representations to ensure their hierarchical properties. Then, we adapt a GNN with novel multi-aspect message-passing-receiving mechanism to capture different influences among users. Next, to characterize the multi-aspect interactions of users on items, we propose an adaptive hyperbolic metric learning method by introducing learnable interactive relations among different aspects. Finally, we utilize the hyperbolic translational distance to measure the plausibility in each user-item pair for recommendation. Experimental results on two public datasets clearly demonstrate that our HyperSoRec not only achieves significant improvement for recommendation performance but also shows better representation ability in hyperbolic space with strong robustness and reliability.